# MATHEMATICAL FLUID MECHANICS - 2020/1

Module code: MAT3041

## Module Overview

This module introduces the ideas of viscous fluid flows which build on the ideas some students would have seen in MAT2050. By the end of the module, students should be familiar with the Navier-Stokes equations, and should be able to solve these equations in under various simplified situations as well as in a variety of geometries.

### Module provider

Mathematics

TURNER Matthew (Maths)

### Module cap (Maximum number of students): N/A

Independent Learning Hours: 117

Lecture Hours: 33

Semester 1

None

## Module content

Introduction.  Definition of a fluid and examples of situations where fluids can be modeled.

Axisymmetric Inviscid Flows. Overview of MAT2050, General solution of Laplace’s equation in spherical geometry, Flow around a sphere, Flow associated with a singing bubble.

The Navier-Stokes Equations. Stress/Strain relation, Derivation of Navier-Stokes equations, Boundary conditions, Dynamical Similarity,

Vorticity dynamics. Derivation of vorticity equation, Physical interpretation, Burger's vortex.

Exact solutions of the Navier-Stokes Equations. 2D flow between plane parallel walls (steady/unsteady), Oscillating plate, Flow in rectangular channel, Pipe flow, Flow between rotating cylinders, The stirring problem, Unsteady line vortex.

Mathematical Boundary Layers. Asymptotic theory for algebraic equations, Matched asymptotic expansions.

Fluid Boundary layer Theory. Derivation of boundary layer equations, Blasius boundary layer, Falkner-Skan solutions.

Application of Boundary Layer Theory: Jets, Wakes

Very Viscous Flow. Lubrication theory, Viscous flow past a sphere.

## Assessment pattern

Assessment type Unit of assessment Weighting
School-timetabled exam/test In-Semester Test (50 minutes) 20
Examination Examination: 2 hour written examination 80

N/A

## Assessment Strategy

The assessment strategy is designed to provide students with the opportunity to demonstrate:

• Understanding of the methods required to solve complex fluid flow problems.

• Subject knowledge through the recall of definitions as well as explaining why certain simplifications to the velocity field can be made and under what conditions certain approximations breakdown, also through a physical understanding of fluid problems.

• Analytic ability through the solution of unseen and seen similar problems in the test and exam.

Thus, the summative assessment for this module consists of:

• One two hour examination (three of four best answers contribute to exam mark) at the end of the Semester; worth 80% module mark.

• One in-semester test; worth 20% module mark.

Formative assessment and feedback

Students receive written feedback via three marked, but unassessed, coursework assignments over an 11 week period.  In addition, verbal feedback is provided by lecturer in lectures and in office hours.

## Module aims

• Introduce students to viscous fluids in various simple geometries, and with various boundary conditions.
• Enable students to solve the Navier-Stokes equations in simple situations.
• Illustrate how fluid mechanics is connected to various problems in the real world, such as in engineering, and how the techniques learnt in this course can be applied to these problems.

## Learning outcomes

 Attributes Developed 001 Solve inviscid fluid problems in spherical polar geometry such as finding the frequency of an oscillating bubble or calculating the Stokes streamfunction given an axisymmetric velocity potential KCT 002 Discuss the role of viscosity in a fluid, and to be able to calculate the viscous stress on a solid surface given the stress tensor. KC 003 Generate exact solutions to the Navier-Stokes equations in both Cartesian and cylindrical polar coordinates. KC 004 Non-dimensionalize the Navier-Stokes equations with and without the effect of gravity and define the Reynolds number and Froude number. Also to demonstrate an understanding of the concept of dynamical similarity. KCT 005 Calculate solutions to algebraic equations and simple ODEs which contain a small parameter or a boundary layer, and demonstrate an understanding of when these approximations breakdown. KCT 006 Apply scale analysis to rescale the Navier-Stokes equations and obtain the boundary layer equations. Also to be able to use these equations to form solutions to problems involving jets and wakes. KC 007 Apply scale analysis to derive the thin film equations from the Navier-Stokes equations and generate solutions of these equations. KC

Attributes Developed

C - Cognitive/analytical

K - Subject knowledge

T - Transferable skills

P - Professional/Practical skills

## Methods of Teaching / Learning

The learning and teaching strategy is designed to provide:

• A thorough account of exact solutions to the Navier-Stokes equations in a variety of geometries and under various simplifications.

• Experience (through demonstration) of the methods and techniques used to solve problems in fluid mechanics.

The learning and teaching methods include:

Teaching will be by lectures and problem classes. Students will receive partial notes with the missing sections filled in during lectures. In addition to reading the lecture notes, students will learn by tackling a wide range of problems. Students are strongly encouraged to use the books listed as background reading on the subject.

Three hours per week (lectures and problem classes) over an 11 week period.

Indicated Lecture Hours (which may also include seminars, tutorials, workshops and other contact time) are approximate and may include in-class tests where one or more of these are an assessment on the module. In-class tests are scheduled/organised separately to taught content and will be published on to student personal timetables, where they apply to taken modules, as soon as they are finalised by central administration. This will usually be after the initial publication of the teaching timetable for the relevant semester.

Upon accessing the reading list, please search for the module using the module code: MAT3041

## Programmes this module appears in

Programme Semester Classification Qualifying conditions
Mathematics MSc 1 Optional A weighted aggregate mark of 40% is required to pass the module
Mathematics with Music BSc (Hons) 1 Optional A weighted aggregate mark of 40% is required to pass the module
Physics BSc (Hons) 1 Optional A weighted aggregate mark of 40% is required to pass the module
Mathematics MMath 1 Optional A weighted aggregate mark of 40% is required to pass the module
Mathematics with Statistics BSc (Hons) 1 Optional A weighted aggregate mark of 40% is required to pass the module
Mathematics BSc (Hons) 1 Optional A weighted aggregate mark of 40% is required to pass the module
Mathematics and Physics BSc (Hons) 1 Optional A weighted aggregate mark of 40% is required to pass the module
Mathematics and Physics MPhys 1 Optional A weighted aggregate mark of 40% is required to pass the module
Mathematics and Physics MMath 1 Optional A weighted aggregate mark of 40% is required to pass the module

Please note that the information detailed within this record is accurate at the time of publishing and may be subject to change. This record contains information for the most up to date version of the programme / module for the 2020/1 academic year.